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Thermodynamics of Perfect Fluids from Scalar Field Theory

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Thermodynamics of Perfect Fluids from Scalar Field Theory PHYSICAL REVIEW D 94, 025034 (2016) Thermodynamics of perfect fluids from scalar field theory † ‡ Guillermo Ballesteros,1,2,* Denis Comelli,3, and Luigi Pilo4,5, 1Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS91191 Gif-sur-Yvette, France 2CERN, Theory Division, 1211 Geneva, Switzerland 3INFN, Sezione di Ferrara, 44124 Ferrara, Italy 4Dipartimento di Scienze Fisiche e Chimiche, Università di L’Aquila, I-67010 L’Aquila, Italy 5INFN, Laboratori Nazionali del Gran Sasso, I-67010 Assergi, Italy (Received 1 June 2016; published 27 July 2016) The low-energy dynamics of relativistic continuous media is given by a shift-symmetric effective theory of four scalar fields. These scalars describe the embedding in spacetime of the medium and play the role of Stückelberg fields for spontaneously broken spatial and time translations. Perfect fluids are selected imposing a stronger symmetry group or reducing the field content to a single scalar. We explore the relation between the field theory description of perfect fluids to thermodynamics. By drawing the correspondence between the allowed operators at leading order in derivatives and the thermodynamic variables, we find that a complete thermodynamic picture requires the four Stückelberg fields. We show that thermodynamic stability plus the null-energy condition imply dynamical stability. We also argue that a consistent thermodynamic interpretation is not possible if any of the shift symmetries is explicitly broken. DOI: 10.1103/PhysRevD.94.025034 I. INTRODUCTION effective field theory (EFT) of continuous media [4,5], which turns to have an ample range of applications. In order Fluid dynamics and thermodynamics are probably the to describe continuous media beyond anisotropic elastic oldest and better known examples of effective descriptions solids, the field content of the pull-back formalism must be of a complicated underlying system in terms of a small extended with a fourth scalar [5]. This allows to include number of macroscopic variables. Systems that admit a superfluids in the picture and also more complex objects fluid description are found in nature at widely separate that are not (yet, maybe) found in nature, such as super- distance scales and energy regimes: from cosmological and solids; see also [6] for other possible types of media. The astrophysical applications to heavy-ion physics and non- fourth scalar can be interpreted as the carrier of an extra relativistic condensed matter. A convenient formulation Uð1Þ charge [5] or as an internal time coordinate of the self- of fluid dynamics in the nondissipative limit is the pull- gravitating medium [7], offering a suggestive link to back formalism—see [1] for a review—where a fluid is massive gravity theories and, in general, models of modi- described through an ensemble of three derivatively fied gravity; see [4,7] and references therein. coupled scalars that are interpreted as comoving coordi- In this work, we focus on perfect fluids, which corre- nates of the fluid’s elements. Within this formalism, fluid spond to two specific subclasses of the EFT of continuous dynamics can be derived from an unconstrained action media at leading order in derivatives (LO), as Fig. 1 principle. A related approach was developed separately illustrates. Although these systems can be considered the to obtain a field theory, symmetry driven, description of the fluctuations—sound waves—propagating in fluids and other types of continuous nonrelativistic media, see [2,3]. The relevant degrees of freedom from this point of view, which we can call phonons, can be identified with the Goldstone bosons of spontaneously broken translational symmetries in the pull-back formalism. Given this, the two approaches can be blended together into a fully relativistic *[email protected] † [email protected] ‡ [email protected] FIG. 1. Red continuous arrows represent the symmetries (4.2) Published by the American Physical Society under the terms of and (4.7) leading to perfect fluids at leading order in derivatives in the Creative Commons Attribution 3.0 License. Further distri- the EFT of nondissipative continuous media. The blue dashed bution of this work must maintain attribution to the author(s) and arrow indicates that restricting the field content to a single the published article’s title, journal citation, and DOI. (temporal) Stückelberg field leads to (irrotational) perfect fluids. 2470-0010=2016=94(2)=025034(15) 025034-1 Published by the American Physical Society BALLESTEROS, COMELLI, and PILO PHYSICAL REVIEW D 94, 025034 (2016) simplest ones at the level of the energy-momentum tensor, the volume. A simple scaling argument shows that they are not free of subtleties [5], and there is an ongoing s ¼ S=V, the entropy density, is a function of the energy effort towards understanding their properties in depth. Here density, ρ ¼ E=V, and the particle number density, we build upon the work of [5]—see also [4]—where the n ¼ N=V; namely,1 thermodynamic interpretation of effective perfect fluids was studied. We extend the thermodynamic correspond- s ¼ sðρ;nÞ; ð2:1Þ ences proposed in [4,5], obtaining a thermodynamic dic- tionary that we have condensed in Table II. which constitutes the fundamental relation containing Our analysis leads to the conclusion that a general all the thermodynamic information of any simple thermodynamic picture (away from specific limits) requires system. Expressing this relation in the equivalent energy indeed four scalar fields (instead of just three) and, representation, consequently, implies an extension of the pull-back for- malism. Remarkably, the form of the action required for ρ ¼ ρðs; nÞ; ð2:2Þ such a thermodynamic picture is determined by a symmetry group that constitutes a specific set of continuous field and taking its differential, we get the first principle of redefinitions, selecting just two effective operators [5]. thermodynamics: We show that a consistent thermodynamic interpretation requires, in any case, a shift symmetry for each field in the dρ ¼ Tds þ μdn; ð2:3Þ effective action. This is interesting because such symmetry is precisely the minimal requirement to have an EFT where the temperature and the chemical potential are organized as a derivative expansion; see [4,5,8,9]. defined as Moreover, shift symmetries are essential for the under- standing of phonons—the degrees of freedom responsible ∂ρ ∂ρ ≡ μ ≡ for the propagation of sound–as Goldstone bosons [2,3] T ∂ ; ∂ : ð2:4Þ s n n s and also [10]. Finally, we argue that thermodynamic stability of perfect From the additivity of the energy E and the entropy S, the fluids plus the null-energy condition guarantee dynamical Euler relation follows, stability, i.e. the absence of ghost degrees of freedom and of exponential growth of fluctuations (around Minkowski ρ þ p ¼ Ts þ μn; ð2:5Þ spacetime). This holds true for all the types of perfect fluids allowed by the EFT. where p is the intensive variable pressure: Whereas the existence of an effective action description of nondissipative fluid dynamics should not be surprising, ∂E p ¼ − : ð2:6Þ the fact that this action leads to a complete thermodynamic ∂V description of perfect fluids is remarkable and far reaching. S;N Having a unified and general relativistic description of The differential of the Euler relation, together with the first nondissipative dynamics and thermodynamics at the action principle, leads to the fact the intensive variables p, T and μ level may open the possibility for novel applications of the are not independent but satisfy the Gibbs-Duhem relation: pull-back formalism and the effective theory of fluids. dp ¼ sdT þ ndμ: ð2:7Þ II. THERMODYNAMICS IN A NUTSHELL Thermodynamics assumes that the equilibrium states of Given two equations among (2.3), (2.5), and (2.7), the third a simple system can be entirely characterized by the follows. extensive variables volume, V, energy, E, and particle From the definitions of the intensive variables p, T and μ, three equations of state can be written in the energy number of each species, Ni. In addition, it postulates the existence of a function of the extensive variables: the representation: entropy, S, which is maximized in the evolution of the system. These two assumptions constitute the first and T ¼ Tðs; nÞ;p¼ pðs; nÞ; μ ¼ μðs; nÞ: ð2:8Þ second postulates of thermodynamics. The third and fourth postulates establish, respectively, that in a composite All together, these three equations of state are equivalent to system the entropy is an additive function over the the fundamental relation (2.2) or (2.1). Keeping only one or constituent subsystems and that the entropy vanishes at two of them among the three leads to some information zero temperature [11]. For our purposes, it is convenient to use intensive 1In relativistic hydrodynamics, n is usually meant to represent variables, defined by dividing the extensive variables over a charge density. 025034-2 THERMODYNAMICS OF PERFECT FLUIDS FROM SCALAR … PHYSICAL REVIEW D 94, 025034 (2016) TABLE I. Thermodynamic potentials and variables. Thermodynamic potential Independent variables Legendre transf. to ρ Conjugate variables Energy density ρ s, n none ∂ρ μ ∂ρ T ¼ ∂s jn, ¼ ∂n js Free-energy density F T, n F ρ − Ts − ∂F μ ∂F ¼ s ¼ ∂T jn, ¼ ∂n jT Grand potential density ω T, μω−p ρ − Ts − μn − ∂ω − ∂ω ¼ ¼ s ¼ ∂T jμ, n ¼ ∂μ jT Potential density I s, μ I ρ − μn ∂I − ∂I ¼ T ¼ ∂s jμ, n ¼ ∂μ js about the system. Clearly, the equations of state can also be fluid, the entropy and the particle number currents are both expressed in different ways depending on the two inde- parallel to vμ, and so we write pendent variables that are chosen. In general, given a simple system, two thermodynamic variables among nμ ¼ nvμ;sμ ¼ svμ: ð3:3Þ fs; T; n; μ; ρ;pg can be taken as independent.
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